We consider two models for branched transport: the one introduced in Bernot et al. (Publ Mat 49:417-451, 2005), which makes use of
a functional defined on measures over the space of Lipschitz paths, and the path
functional model presented in Brancolini et al. (J Eur Math Soc 8:415-434, 2006), where one minimizes some suitable action functional defined
over the space of measure-valued Lipschitz curves, getting sort of a Riemannian metric on the space of probabilities, favouring atomic measures, with a cost depending on the masses of each of their atoms.
We prove that modifying the latter model according to Brasco (Ann Mat Pura Appl 189:95-125, 2010), then the two models turn out to be equivalent.